DARGAN M. W. FRIERSON UNIVERSITY OF WASHINGTON, DEPARTMENT OF ATMOSPHERIC SCIENCES WEEK 8 SLIDES GFD II: Balance Dynamics ATM S 542Eady Model Growth rates (imaginary part of frequency) Wave speeds (real part of frequency divided by k) Stable for large wavenumbers, unstable for small wavenumbers Phase speed for unstable modes = mean flow speed at midtroposphere.Eady vs Barotropic Instability Eady model Barotropic instabilityEady vs. Barotropic Instability ! Both require the following conditions on edge waves: ¡ Phase speed matching ¡ Amplitude requirement ÷ Waves have to affect each other ÷ Depends on their reach vs their separation ¡ Phase tilt requirement ÷ Must be so they reinforce each others perturbations ! Eady is 3-D though! Barotropic is only 2-DEady Model of Baroclinic Instability Height (contours) & theta (colors) Vorticity (contours) & w (colors)Heat fluxes Growing mode has heat flux poleward Decaying mode has equatorward heat flux (upgradient!)Eady phase tilts (most unstable mode) Streamfunction Buoyancy Meridional velocity ψb =∂ψ∂zv =∂ψ∂xNote v and b tend to be correlated, i.e., there’s a poleward heat fluxHeat fluxes Edge wavesIdealized CyclogenesisNon-QG Effects Nonlinear baroclinic instability simulations with a QG model and with a primitive equations modelObserved CyclogenesisAlternatives to Eady ! Other important models of baroclinic instability where linear analysis is useful: ¡ Two-layer QG model (“Phillips model”): see Vallis Section 6.6 ¡ Charney model: see Vallis Section 6.9.1Two layer model w/o beta 2 layer problem closely resembles Eady qualitatively Quantitative approach to Eady growth rates as layers are addedTwo layer (Phillips) model with betaTwo layer model with betaPhillips model summary ! Two-layer baroclinic instability: ¡ Still have interacting waves leading to instability ¡ Can qualitatively reproduce Eady results with beta = 0 ¡ With beta, have: ÷ Critical shear for instability ÷ Longwave cutoff ÷ Shortwave cutoffCharney model ! Setup: ¡ Unbounded in vertical, density decreasing with height ¡ Constant shear ¡ Beta effect included ! Mechanism: ¡ Edge wave at surface interacts with a Rossby wave in the troposphereSummary: Extensions to Eady ! Beta: ¡ How to add: ÷ 2-layer model allows for easy analysis of this effect. ÷ Charney model: beta is key in this model ÷ Can also run Eady model numerically with beta. ¡ Results: ÷ In 2-layer model, stabilizes everywhere, causing weaker instability and stabilizing growth rates. This happens preferentially at larger scales. ÷ In Charney model, larger beta pushes shallow modes downward. ÷ In Eady model plus beta, pushes modes downward.Summary: Extensions to Eady ! No upper boundary: ¡ How to add: ÷ Eady model without an upper boundary cannot produce baroclinic instability ÷ With beta, this is the Charney model ¡ Results: ÷ Charney model has baroclinic instability (provided there’s beta) ¢ Deep modes have depth set by density scale height, and are similar to Eady model ¢ Shallow modes also exist though: Rossby waves in interior locking to surface edge waves ¢ Presence of shallow modes imply no short wave cutoff in Charney modelSummary: Extensions to Eady ! Non-uniformity in y ¡ How to add: ÷ Make a localized jet. Then can test linear stability of this, run nonlinear simulations,